On the greatest prime factor of ab+1

We improve some results on the size of the greatest prime factor of integers of the form ab+1, where a and b belong to finite sets of integers with rather large density.Comment: 38 pages. Theorem 3 of this version is improved. One bibliographical item is adde

On the conductor of cohomological transforms

In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.Comment: v2; 41 pages, with important simplifications as well as a number of correction

Algebraic twists of modular forms and Hecke orbits

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ell-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.Comment: v5, final version to appear in GAF

Number of integers represented by families of binary forms

We extend our previous results on the number of integers which are values of some cyclotomic form of degree larger than a given value (see \cite{FW1}), to more general families of binary forms with integer coefficients. Our main ingredient is an asymptotic upper bound for the cardinality of the set of values which are common to two non isomorphic binary forms of degree greater than $3$. We apply our results to some typical examples of families of binary forms.Comment: This revised version is accepted for publication in Acta Arithmetic